Author(s):  
Tim Button ◽  
Sean Walsh

This chapter explores Leibniz's principle of the Identity of Indiscernibles. Model theory supplies us with the resources to distinguish between many different notions of indiscernibility; we can vary: (a) the primitive ideology (b) the background logic and (c) the grade of discernibility. We use these distinctions to discuss the possibility of singling-out “indiscernibles”. And we then use these to distinctions to explicate Leibniz's famous principle. While model theory allows us to make this principle precise, the sheer number of different precise versions of this principle made available by model theory can serve to mitigate some of the initial excitement of this principle. We round out the chapter with two technical topics: indiscernibility in infinitary logic, and the relation between indiscernibility, orders, and stability.


1972 ◽  
Vol 37 (4) ◽  
pp. 677-682 ◽  
Author(s):  
George Metakides

Let α be a limit ordinal with the property that any “recursive” function whose domain is a proper initial segment of α has its range bounded by α. α is then called admissible (in a sense to be made precise later) and a recursion theory can be developed on it (α-recursion theory) by providing the generalized notions of α-recursively enumerable, α-recursive and α-finite. Takeuti [12] was the first to study recursive functions of ordinals, the subject owing its further development to Kripke [7], Platek [8], Kreisel [6], and Sacks [9].Infinitary logic on the other hand (i.e., the study of languages which allow expressions of infinite length) was quite extensively studied by Scott [11], Tarski, Kreisel, Karp [5] and others. Kreisel suggested in the late '50's that these languages (even which allows countable expressions but only finite quantification) were too large and that one should only allow expressions which are, in some generalized sense, finite. This made the application of generalized recursion theory to the logic of infinitary languages appear natural. In 1967 Barwise [1] was the first to present a complete formalization of the restriction of to an admissible fragment (A a countable admissible set) and to prove that completeness and compactness hold for it. [2] is an excellent reference for a detailed exposition of admissible languages.


2018 ◽  
Vol 28 (6) ◽  
pp. 1275-1292
Author(s):  
Antonio Di Nola ◽  
Serafina Lapenta ◽  
Ioana LeuŞtean

2003 ◽  
Vol 68 (1) ◽  
pp. 65-131 ◽  
Author(s):  
Andreas Blass ◽  
Yuri Gurevich

AbstractThis paper developed from Shelah's proof of a zero-one law for the complexity class “choiceless polynomial time,” defined by Shelah and the authors. We present a detailed proof of Shelah's result for graphs, and describe the extent of its generalizability to other sorts of structures. The extension axioms, which form the basis for earlier zero-one laws (for first-order logic, fixed-point logic, and finite-variable infinitary logic) are inadequate in the case of choiceless polynomial time; they must be replaced by what we call the strong extension axioms. We present an extensive discussion of these axioms and their role both in the zero-one law and in general.


1982 ◽  
Vol 87 ◽  
pp. 1-15
Author(s):  
Chiharu Mizutani

Svenonius’ definability theorem and its generalizations to the infinitary logic Lω1ω or to a second order logic with countable conjunctions and disjunctions have been studied by Kochen [1], Motohashi [2], [3] or Harnik and Makkai [4], independently. In this paper, we consider a (Svenonius-type) definability theorem for the intuitionistic predicate logic IL with equality.


1969 ◽  
Vol 34 (2) ◽  
pp. 226-252 ◽  
Author(s):  
Jon Barwise

In recent years much effort has gone into the study of languages which strengthen the classical first-order predicate calculus in various ways. This effort has been motivated by the desire to find a language which is(I) strong enough to express interesting properties not expressible by the classical language, but(II) still simple enough to yield interesting general results. Languages investigated include second-order logic, weak second-order logic, ω-logic, languages with generalized quantifiers, and infinitary logic.


Sign in / Sign up

Export Citation Format

Share Document